In MRI imaging, a subject imaged is placed in a spatially and temporally varying magnetic field so that an imaging nucleus in the subject precesses around the magnetic field with a Larmor frequency .omega.(r, t)=.gamma.B(r, t), where .gamma. is the gyromagnetic moment of the imaging nucleus and B is the magnitude of the magnetic field at time t and at a point r where the imaging nucleus is located. The dependence of the Larmor frequency on position is used to evaluate the spin density, .rho.(x,y,z), of the imaging nuclei as a function of position in the imaged subject. Since .rho.(x,y,z) is a function of the internal features and structures of the imaged subject these features and structures can be visualized.
The magnetic field B (r, t) is generally of the form B.sub.0 +G.sub.x x+G.sub.y y+G.sub.z z in a direction conventionally labeled as the z direction. B.sub.0 is the magnitude of a large and constant homogeneous magnetic field parallel to the z direction, and G.sub.x, G.sub.y and G.sub.z are the x,y, and z gradients of gradient fields, also parallel to the z axis, added to B.sub.0., The time dependence of B (r, t) is a function of the rise times of the gradients G.sub.x, G.sub.y, and G.sub.z and the timing sequence which governs when they are turned on and off.
A signal is elicited form the imaging nuclei by rotating their spins away from the z axis. This creates a component of spin density .rho.(x,y,z) and associated magnetization density, m(x,y,z)=.gamma..rho.(x,y,z) which rotate perpendicular to the z axis with a frequency equal to the Larmor frequency of the imaging nuclei. The rotating magnetic density produces a signal SIG(t) which is sensed by an appropriate receiving antenna.
The relationship between SIG(t) and the spin density .rho.(x,y,z) of the imaging nucleus can be written as SIG(t)=.intg..intg..intg..rho.(x,y,z)exp[i.gamma.(B.sub.0 +G.sub.x x+G.sub.y y+G.sub.z z)t] dxdydz=exp[i.omega..sub.0 t]S(t), where .omega..sub.0 =.gamma.B.sub.0 is the Larmor frequency of the imaging nucleus in B.sub.0 and S(t) is the part of the integral dependent upon .rho.(x,y,z). In the relationship, the relaxation of the magnitude of the spin density in the transverse plane to zero and the recovery of the spin density equilibrium value along the z axis is ignored, and it has been assumed that the angle of rotation of .rho.(x,y,z) away from the z axis is .pi./2.
By changing variables so that k.sub.x =.gamma.G.sub.x t, k.sub.y =.gamma.G.sub.y t and k.sub.z =.gamma.G.sub.z t, S(t) can be written as a function of position in a "k-space". S(t).fwdarw.S(k.sub.x,k.sub.y,k.sub.z)=S(k), and: EQU S(k)=.intg..intg..intg..rho.(x,y,z)exp[i(k.sub.x x+k.sub.y y+k.sub.z z)]dxdydz=.intg..intg..intg..rho.(r)exp[ik.multidot.r]d.sup.3 r.
This last integral is the Fourier transform of the spin density function .rho.(x,y,z) of the imaging nucleus. S(k) is a "k-transform" of .rho.(x,y,z) and k and r are conjugate variables so that .rho.(r)=.intg..intg..intg.S(k)exp[ik.multidot.r]d.sup.3 k.
When the imaging nuclei are first flipped away from the z axis they precess together coherently with a net spin density and magnetization density in the xy plane. With time, however, the coherence in the transverse xy plane decays to zero and the spin density relaxes to the equilibrium state where the imaging nuclei are polarized along the z axis. The decay of net transverse spin density and return to equilibrium along the z axis are characterized by different time constants known as T.sub.2 and T.sub.1 respectively. When inhomogenities in the magnetic field are present, the decay of transverse spin density is accelerated and is characterized by a time constant known as T.sub.2 *. The relaxation times are related by the inequality T.sub.2 *&lt;T.sub.2 &lt;T.sub.1.
Many different techniques have been developed for MRI imaging. All involve procedures for acquiring values for the k-transform, S(k), of a subject imaged at many points, hereafter "read-points", in a raster of points in k-space so that the Fourier transform of S(k) results in a proper evaluation of .rho.(r).
Generally, the problem of evaluating S(k) over three dimensions in k-space is reduced to a two dimensional one. The k-transform S(k) is evaluated for a thin slice perpendicular to the z axis of the subject being imaged, so that S(k).fwdarw.S(k.sub.x,k.sub.y,Z.sub.s) where Z.sub.s is the constant z coordinate of the slice. A three dimensional image is built up from the two dimensional images of many adjacent thin slices acquired for a range of values of Z.sub.s.
Often, data for a k-transform S(k), is acquired using different MRI imaging techniques or operating conditions in different areas of k-space. Ideally the values of data acquired should be independent of data acquisition method. Sets of data acquired for a same k-transform S(k) using different techniques or operating conditions therefore should be consistent with each other. Consistency requires that, at boundaries between two areas in k-space where data for a k-transform is acquired in one of the k-space areas using a technique or conditions different from the technique or conditions used to acquire data for the same k-transform in the other k-space area, the data on either side of the boundaries approach the same values for points on the boundaries i.e. the data must be continuous at the boundaries.
If the data is not consistent, and has discontinuities at boundaries, the discontinuities cause artifacts such as ghosting or ringing. The scale and seriousness of the artifacts is an increasing function of the slope and magnitude of the discontinuities. These seriously degrade an MRI image constructed from the k-transform and generally, solutions are needed to remove or moderate them.
Many discontinuities and sources of discontinuities are removed by standard normalization and calibration procedures. These procedures remove from the data equipment biases and many types of timing errors. Additional corrections to the data are made by dividing out the T.sub.1 and T.sub.2, decay envelope from the data. Finally, data is corrected for T.sub.2 * effects arising from chemical shift and field inhomogenities.
Chemical shift and field inhomogeneity often lead to large discontinuous phase differences between data on opposite sides of a boundary between data sets acquired using different MRI techniques or operating conditions. This occurs when data on opposite sides of the boundary are acquired at significantly different echo times T.sub.E. The phase differences are proportional to the time difference between the echo times and the magnitudes of the chemical shift fields and the field inhomogenities.
One way to remove chemical shift and field inhomogeneity effects is to acquire all data at the same T.sub.E time. This is not possible except with the standard MRI spin echo technique. This technique however is slow and therefore not suitable for many procedures. The new fast imaging techniques must contend with chemical and field inhomogeneity phase effects.
It is possible to consider mathematically removing these effects. Removing the chemical shift and field inhomogeneity effects from the data mathematically requires the acquisition of data additional to the basic k-transform data, such as a magnetic field homogeneity map. This is often impractical or not possible.
Often chemical shift and field inhomogeneity effects are not removed from the data but are prevented from accumulating and concentrating along the boundary. In this way any discontinuities that they might cause are significantly reduced.
This is generally done in prior art by a technique called echo time shifting. In echo time shifting, the T.sub.E times of echo pulses and their associated read gradients in an MRI imaging sequence are shifted so as to prevent a large difference in echo time T.sub.E from appearing between k-transform data acquired at adjacent read points. Since the phase difference between data points due to chemical shift and field inhomogeneity is proportional to the difference between the T.sub.E times at which they were taken, this approach prevents any large discontinuous phase shifts caused by chemical shift and field inhomogeneity from occurring in the k-transform data.
For example, assume a phase discontinuity occurs in the data of a k-transform that causes severe ghosting or ringing in the image constructed from the k-transform. Assume that the phase shift is caused by chemical shift and field inhomogeneity, and that it occurs across a boundary common to a first and a second data subset of a k-transform. Let .phi., be the phase difference and assume it results from an echo time difference .DELTA.T.sub.E, between the echo time at which the data at the boundary in the first data set is acquired and the echo time at which the data at the boundary for the second data set is acquired. Further assume that the data in the first data set is acquired using n read gradients, with the data at the boundary from the first data set coming from the n-th read gradient.
To prevent or reduce this discontinuity, the time interval between the n consecutive read gradients used to accumulate data in the first data set is increased by .delta.T.sub.E =.DELTA.T.sub.E /n. As a result, the second of the n consecutive read gradients is shifted in time by .delta.T.sub.E, the third by 2.delta.T.sub.E, the fourth by 3.delta.T.sub.E, and so on, with the n-th read gradient shifted by (n-1) .delta.T.sub.E. The chemical and field inhomogeneity phase difference between each of the n read gradients in the first data set is increased by .phi./n and the phase difference at the boundary between the data sets is decreased by a factor of n to .phi./n. The phase difference has not been removed, it has been redistributed over the data by echo time shifting. The phase difference at the boundary has been decreased by a factor of n at the expense of increasing the phase difference between data taken from consecutive read gradients used for the first data set by .phi./n. The discontinuity at the boundary has been substantially removed and if phase differences on the order of .phi./n do not cause ghosting or ringing the image reconstructed from the k-transform is improved.
The problem of discontinuities in data at boundaries often arises in procedures which use echo planar imaging sequences (EPI) to acquire k-transform data. EPI sequences generally require very large, rapidly changing gradient fields and wide band receivers. These fields and their time derivatives may exceed limitations established to prevent their causing undesirable biological effects. Additionally, the sequence is generally very strongly affected by T.sub.2 * decay, so that as the sequence progresses, the signal to noise ratio decreases. Both of these problems are moderated by decreasing the duration of the sequence. As a result, when data for a k-transform of a subject is acquired in a k-space using EPI, the data is often acquired using a set of short EPI sequences instead of one single long EPI sequence. The k-space is partitioned into different parts, hereafter "read sectors", and each shortened EPI sequence acquires data in a different read sector. A boundary is created where one read sector ends and another begins.
In order to combine the data from these shortened EPI sequences into one consistent data set, the data of a k-transform must be continuous across read sector boundaries. This requires that the data for each read sector be calibrated and normalized and corrected for T.sub.2 decay and chemical shift and field inhomogeneity.
GRASE imaging is another MRI imaging technique that creates boundaries between sets of data acquired for a same k-transform S(k.sub.x,k.sub.y), of a subject imaged. A GRASE imaging sequence employs a 90.degree. RF slice selection pulse followed by N GRASE cycles, where each GRASE cycle comprises a 180.degree. RF spin focusing pulse and associated selection gradient pulse, followed by M read gradient pulses with alternating polarity, during which data is acquired. For each GRASE cycle for S(k.sub.x,k.sub.y) data is acquired for M complete read lines. Phase encoding to determine the k.sub.y coordinate of read lines scanned in a GRASE cycle is accomplished with a series of variable magnitude phase gradient pulses, generally coincident with the zero crossing points between read gradient pulses. A single, complete GRASE
Thus, the method of removing k-space discontinuities of the patent of Liu et al fails to provide the following elements of the claims of the application:
determining a width for a transition zone in the k-space at and including the boundary; PA1 calculating bridging data from data in the overlap data set for k-coordinates in the transition zone; and PA1 replacing data at k-coordinates in the transition zone with bridging data, discusses GRASE imaging and illustrates the way in which GRASE imaging divides k-space into different data acquisition areas.
All the data acquired in a particular read sector is generally acquired from a same read gradient pulse of repeated GRASE cycles, e.g. all the data for read points in the third read sector are acquired from the third read gradient of repeated GRASE cycles. As a result, the data in each read sector are all acquired at a same unique echo time T.sub.E, which is different from the T.sub.E at which data is acquired for any of the other read sectors . There is therefore, no difference in phase from chemical shift and field inhomogeneity between different data belonging to a same read sector, but there is a difference in phase arising from chemical shift and field inhomogeneity between data belonging to different read sectors. Each read sector has its own unique chemical shift and field inhomogeneity phase offset which is different from the chemical shift and field inhomogeneity phase offset of other read sectors. Therefore, the data in each read sector exhibits T.sub.2 decay but not T.sub.2 * decay while the data from different read sectors taken with a same GRASE cycle exhibit T.sub.2 * decay.
In order for the data from the different read sectors to be combined into a single consistent set of data continuous at the boundaries between read sectors the data in each read sector has to be normalized, corrected for T.sub.2 decay, and for chemical shift and field inhomogeneity.
In both examples even after the appropriate normalization, calibration and correction procedures are carried out, the data at boundaries may sometimes still exhibit discontinuities in phase and or in magnitude. In addition, echo time shifting techniques for preventing or ameliorating discontinuities arising from chemical shifts and field inhomogenities require special software and hardware that is expensive and frequently not available.
It would therefore be desirable to have a simple mathematical procedure to remove or reduce discontinuities that occur at boundaries between data subsets of a k-transform that are caused by chemical shifts and field inhomogenities and that persist after standard normalization, calibration and T.sub.2 decay corrections on the data are carried out.